Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1205.0885 · The Journal of Chemical Physics

Ehrenfest dynamics is purity non-preserving: A necessary ingredient for decoherence

José L. Alonso, Jesús Clemente-Gallardo, Juan C. Cuchí, P. M. Echenique, Fernando Falceto

Published Aug 2, 2012• 26 citations
arXivHTMLPDF

Abstract

We discuss the evolution of purity in mixed quantum/classical approaches to electronic nonadiabatic dynamics in the context of the Ehrenfest model. As it is impossible to exactly determine initial conditions for a realistic system, we choose to work in the statistical Ehrenfest formalism that we introduced in Alonso et al. [J. Phys. A: Math. Theor. 44, 396004 (2011)]. From it, we develop a new framework to determine exactly the change in the purity of the quantum subsystem along with the evolution of a statistical Ehrenfest system. In a simple case, we verify how and to which extent Ehrenfest statistical dynamics makes a system with more than one classical trajectory, and an initial quantum pure state become a quantum mixed one. We prove this numerically showing how the evolution of purity depends on time, on the dimension of the quantum state space D, and on the number of classical trajectories N of the initial distribution. The results in this work open new perspectives for studying decoherence with Ehrenfest dynamics.

DOI
10.1063/1.4737861

Related papers

Ehrenfest Statistical Dynamics in Chemistry: Study of Decoherence Effects

José Luis Alonso, Pierpaolo Bruscolini, Alberto Castro +3 more

In previous works, we introduced a geometric route to define our Ehrenfest statistical dynamics (ESD) and we proved that, for a simple toy model, the resulting ESD does not preserve purity. We now take a step further: we investigate decoherence and pointer basis in the ESD model by considering some uncertainty in the degrees of freedom of a simple but realistic molecular model, consisting of two classical cores and one quantum electron. The Ehrenfest model is sometimes discarded as a valid approximation to nonadiabatic coupled quantum-classical dynamics because it does not describe the decoherence in the quantum subsystem. However, any rigorous statistical analysis of the Ehrenfest dynamics, such as the described ESD formalism, proves that decoherence exists. In this article, decoherence in ESD is studied by measuring the change in the quantum subsystem purity and by analyzing the appearance of the pointer basis to which the system decoheres, which for our example is composed of the eigenstates of the electronic Hamiltonian.

arXiv:1801.06137•Journal of Chemical Theory and Computation•Jun 26, 2018•10 citations

Ehrenfest statistical dynamics in chemistry: study of decoherence effects and of its equilibrium distribution

José Luis Alonso, Pierpaolo Bruscolini, Alberto Castro +3 more

We investigate decoherence, pointer states and the equilibrium distribution in the Ehrenfest Statistical Dynamics (ESD) model by considering ensembles of trajectories of simple but realistic molecular models, consisting of two classical cores and one quantum electron. The Ehrenfest model is sometimes discarded as a valid approximation to non-adiabatic coupled quantum-classical dynamics because it does not describe the decoherence in the quantum subsystem, which should exist due to its interaction with the classical subsystem. However, any rigorous statistical analysis of the Ehrenfest dynamics, such as the described ESD formalism described, proves that some decoherence-like effects, in particular purity decreasing, exist when ensembles of trajectories are considered. In this article, decoherence in ESD is studied by measuring the change in the quantum subsystem purity and by analysing the appearance of pointer states to which the system decoheres, which for our example are the eigenstates of the electronic Hamiltonian. Furthermore, following the formalism presented in previous works, the true equilibrium distribution for the ESD determined following the Balescu approach is used in the definition of potential energy surfaces dependent on the temperature and in the computation of the temperature dependence of the internuclear distance.

arXiv:1801.06137?context=physics•Jan 18, 2018•0 citations

Quantifying Early Time Quantum Decoherence Dynamics through Fluctuations

Bing Gu, Ignacio Franco

We introduce a general but simple relation between the timescale for quantum coherence loss and the initial fluctuations of operators that couple a quantum system with a surrounding bath. The relation allows the prediction and measurement of early time decoherence dynamics for any open quantum system, through purity, without reconstructing the system's many-body density matrix. It is applied to predict the decoherence time for basic models-the Holstein chain, spin-boson and Caldeira-Legget models-commonly employed to capture electronic, vibrational, and vibronic dynamics in molecules. Such development also offers a practical platform to test the ability of approximate quantum dynamics methods to capture decoherence. In particular, a class of mixed quantum-classical schemes for molecular dynamics where the bath is treated classically, such as Ehrenfest dynamics, are shown to correctly capture short-time decoherence when the initial conditions are sampled from the Wigner distribution. These advances provide a useful platform to develop decoherence times for molecular processes and to test approximate molecular dynamics methods.

The Journal of Physical Chemistry Letters•Aug 19, 2017•43 citations

Electronic decoherence along a single nuclear trajectory

Matisse Wei-Yuan Tu, Eberhard K. U. Gross

We describe an approach to subsystem decoherence without the usual tracing out of the environment. The subsystem of focus is described entirely by a pure state evolving nonunitarily along a single classical trajectory of its environment. The approach is deduced from the exact factorization framework for arbitrary systems of electrons and nuclei. The nonunitarity of the electronic dynamics arises exclusively from nonadiabatic correlations between electrons and nuclei. We demonstrate that the approach correctly describes the coherence gain and the subsequent decoherence for the example of a nuclear trajectory passing through an avoided crossing, the prototypical case where single-trajectory Ehrenfest dynamics fails to produce decoherence.

arXiv:2502.08247•Physical Review Research•Sep 17, 2025•2 citations

Decoherence and Its Role in Electronically Nonadiabatic Dynamics

Yinan Shu, Donald G. Truhlar

Decoherence is the tendency of a time-evolved reduced density matrix for a subsystem to assume a form corresponding to a statistical ensemble of states rather than a coherent combination of pure-state wave functions. When a molecular process involves changes in the electronic state and the coordinates of the nuclei, as in ultraviolet or visible light photochemistry or electronically inelastic collisions, the reduced density matrix of the electronic subsystem suffers decoherence, due to its interaction with the nuclear subsystem. We present the background necessary to conceptualize this decoherence; in particular, we discuss the density matrix description of pure states and mixed states, and we discuss pointer states and decoherence time. We then discuss how decoherence is treated in the coherent switching with decay of mixing algorithm and the trajectory surface hopping method for semiclassical calculations of electronically nonadiabatic processes.

Journal of Chemical Theory and Computation•Jan 9, 2023•54 citations

Semiclassical Ehrenfest paths in open quantum systems

Xiao-Kan Guo

We study the semiclassical Ehrenfest trajectories in open quantum systems. We first give a self-contained derivation of the Fokker--Planck equation that governs the time evolution of the mixing measure for a Gaussian mixture. Then, we embed the generalized Ehrenfest theorem recently obtained for open quantum systems into this phase-space picture to study the time evolution of the expectations of observables with respect to the Gaussian mixture. We show how the coherent and irreversible contributions are separated in the frictionless harmonic regime relevant to the semiclassical approximation. Our work provides a transparent phase-space interpretation of the emergence of classical trajectories in open quantum dynamics.

quant-pharXiv:2604.27328•May 1, 2026•0 citations
We discuss the evolution of purity in mixed quantum/classical approaches to electronic nonadiabatic dynamics in the context of the Ehrenfest model. As it is impossible to exactly determine initial conditions for a realistic system, we choose to work in the statistical Ehrenfest formalism that we introduced in Alonso et al. [J. Phys. A: Math. Theor. 44, 396004 (2011)]. From it, we develop a new framework to determine exactly the change in the purity of the quantum subsystem along with the evolution of a statistical Ehrenfest system. In a simple case, we verify how and to which extent Ehrenfest statistical dynamics makes a system with more than one classical trajectory, and an initial quantum pure state become a quantum mixed one. We prove this numerically showing how the evolution of purity depends on time, on the dimension of the quantum state space D, and on the number of classical trajectories N of the initial distribution. The results in this work open new perspectives for studying decoherence with Ehrenfest dynamics.
In previous works, we introduced a geometric route to define our Ehrenfest statistical dynamics (ESD) and we proved that, for a simple toy model, the resulting ESD does not preserve purity. We now take a step further: we investigate decoherence and pointer basis in the ESD model by considering some uncertainty in the degrees of freedom of a simple but realistic molecular model, consisting of two classical cores and one quantum electron. The Ehrenfest model is sometimes discarded as a valid approximation to nonadiabatic coupled quantum-classical dynamics because it does not describe the decoherence in the quantum subsystem. However, any rigorous statistical analysis of the Ehrenfest dynamics, such as the described ESD formalism, proves that decoherence exists. In this article, decoherence in ESD is studied by measuring the change in the quantum subsystem purity and by analyzing the appearance of the pointer basis to which the system decoheres, which for our example is composed of the eigenstates of the electronic Hamiltonian.
We investigate decoherence, pointer states and the equilibrium distribution in the Ehrenfest Statistical Dynamics (ESD) model by considering ensembles of trajectories of simple but realistic molecular models, consisting of two classical cores and one quantum electron. The Ehrenfest model is sometimes discarded as a valid approximation to non-adiabatic coupled quantum-classical dynamics because it does not describe the decoherence in the quantum subsystem, which should exist due to its interaction with the classical subsystem. However, any rigorous statistical analysis of the Ehrenfest dynamics, such as the described ESD formalism described, proves that some decoherence-like effects, in particular purity decreasing, exist when ensembles of trajectories are considered. In this article, decoherence in ESD is studied by measuring the change in the quantum subsystem purity and by analysing the appearance of pointer states to which the system decoheres, which for our example are the eigenstates of the electronic Hamiltonian. Furthermore, following the formalism presented in previous works, the true equilibrium distribution for the ESD determined following the Balescu approach is used in the definition of potential energy surfaces dependent on the temperature and in the computation of the temperature dependence of the internuclear distance.
We introduce a general but simple relation between the timescale for quantum coherence loss and the initial fluctuations of operators that couple a quantum system with a surrounding bath. The relation allows the prediction and measurement of early time decoherence dynamics for any open quantum system, through purity, without reconstructing the system's many-body density matrix. It is applied to predict the decoherence time for basic models-the Holstein chain, spin-boson and Caldeira-Legget models-commonly employed to capture electronic, vibrational, and vibronic dynamics in molecules. Such development also offers a practical platform to test the ability of approximate quantum dynamics methods to capture decoherence. In particular, a class of mixed quantum-classical schemes for molecular dynamics where the bath is treated classically, such as Ehrenfest dynamics, are shown to correctly capture short-time decoherence when the initial conditions are sampled from the Wigner distribution. These advances provide a useful platform to develop decoherence times for molecular processes and to test approximate molecular dynamics methods.
We describe an approach to subsystem decoherence without the usual tracing out of the environment. The subsystem of focus is described entirely by a pure state evolving nonunitarily along a single classical trajectory of its environment. The approach is deduced from the exact factorization framework for arbitrary systems of electrons and nuclei. The nonunitarity of the electronic dynamics arises exclusively from nonadiabatic correlations between electrons and nuclei. We demonstrate that the approach correctly describes the coherence gain and the subsequent decoherence for the example of a nuclear trajectory passing through an avoided crossing, the prototypical case where single-trajectory Ehrenfest dynamics fails to produce decoherence.
Decoherence is the tendency of a time-evolved reduced density matrix for a subsystem to assume a form corresponding to a statistical ensemble of states rather than a coherent combination of pure-state wave functions. When a molecular process involves changes in the electronic state and the coordinates of the nuclei, as in ultraviolet or visible light photochemistry or electronically inelastic collisions, the reduced density matrix of the electronic subsystem suffers decoherence, due to its interaction with the nuclear subsystem. We present the background necessary to conceptualize this decoherence; in particular, we discuss the density matrix description of pure states and mixed states, and we discuss pointer states and decoherence time. We then discuss how decoherence is treated in the coherent switching with decay of mixing algorithm and the trajectory surface hopping method for semiclassical calculations of electronically nonadiabatic processes.
We study the semiclassical Ehrenfest trajectories in open quantum systems. We first give a self-contained derivation of the Fokker--Planck equation that governs the time evolution of the mixing measure for a Gaussian mixture. Then, we embed the generalized Ehrenfest theorem recently obtained for open quantum systems into this phase-space picture to study the time evolution of the expectations of observables with respect to the Gaussian mixture. We show how the coherent and irreversible contributions are separated in the frictionless harmonic regime relevant to the semiclassical approximation. Our work provides a transparent phase-space interpretation of the emergence of classical trajectories in open quantum dynamics.